Source: On the Range of Cosine Transform of Distributions for Torus-invariant Complex Minkowski Spaces

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Source: THE PROBABILISTIC ESTIMATES ON THE LARGEST AND SMALLEST q-SINGULAR VALUES OF RANDOM MATRICES

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**We know that , for any , can be expressed as an integral on by spreading a measure on it, which is**

** (1)**

**for some constant because , where**

$latex \delta_{\frac{\pi}{4}}(\theta)=\begin{cases}

+\infty, & \theta=\frac{\pi}{4}\\

0, & \theta\ne\frac{\pi}{4}.\end{cases}$ (2)

**which satisfies and**

$latex \delta_{\frac{3\pi}{4}}(\theta)=\begin{cases}

+\infty, & \theta=\frac{3\pi}{4}\\

0, & \theta\ne\frac{3\pi}{4}.\end{cases}$ (3)

**satisfying are modified Dirac delta functions.**

**One might also be able to put a measure on for the complex norm , , which can be written in real norm as . A problem is what function would satisfy**

** (4)**

**if one parametrizes by the Hopf coordinates**

** (5)**

**One can show that the function is actually some function independent of and by the invariance of the complex norm under action, so it can be denoted as . But it is unknown yet what exactly the function is, because the invariance yields that , but this doesn’t produce an appropriate function in the general expression for the complex infinity norm. If one looks back the case of real infinity norm, the way of expressing the infinity norm is by making use of the absolute values on sum and difference smartly to sift out the maximum from two number. But in the complex case the double integral on the torus**

** (6)**

**eliminates the feasibility of sifting out the maximal modulus.**

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**Suppose is a compact -dimensional submanifold with smooth boundary in the plane . From symplectic geometry, the natural symplectic form on can be written as , where . If one computes the integral,**

**(1)**

**where denotes the codisk bundle of , since is an one dimensional curve in , it makes the integral containing measuring the perturbations of base points of cotangent vectors in be zero. Hence each term of the right hand side of (1) is zero. So one has that , which is also true in general when is a compact hypersurface contained in some affine -plane in .**

**By the way, Gowers wrote a post, one way of looking at Cauchy theorem, which is very interesting and tells us that the Cauchy’s theorem is the natural generalization in -dimension of the statement is a constant for , that, we know, is a direct consequence of the second fundamnetal theorem of calculus which is the special case of the Stokes’ theorem in one dimension. So in this sense, Stokes’ theorem is the “generator” of all the theorems and statements of this kind.**

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